The converse Bernstein-coloring conjecture for independence in $\mathcal{B}_{m,n}(d,d)$

Let $G$ be a bipartite graph, let $d\geq 0$ be an integer, and let $\mathcal{B}_{m,n}(d,d)$ denote the matroid introduced in the paper. A $d$-coloring of the edges of $G$ is $d$-Bernstein if it has no monochromatic cycles and there is a labeling $c\colon V(G)\to\mathbb{R}^d$, with $c(v)=(c_1(v),c_2(v),\dots,c_d(v))$, such that

$$c_1(v)+c_2(v)+\cdots+c_d(v)=0$$

for every $v\in V(G)$, and, for every edge $(u,v)\in E(G)$ with color $i$,

$$c_i(u)+c_i(v)>c_j(u)+c_j(v)$$

for every $j\in[d]\setminus\{i\}$. The converse Bernstein-coloring conjecture. If $G$ is independent in $\mathcal{B}_{m,n}(d,d)$, then $G$ admits a $d$-coloring that is $d$-Bernstein. This is presented as the converse to the preceding proposition, which proves that a $d$-Bernstein coloring implies independence in the relevant dual matroid; the supplied text gives no resolution status or further evidence, so the conjecture remains open.